I know that under large signal analysis, a capacitor becomes open, but under small, it becomes a short. My main question is what is an inductor under small signal analysis. Is it a short under large signal, but open under small signal?

The impedance of a capacitor or inductor depends upon the value of said components as compared to the frequency of the AC waveform across them.

DC is (ideally) blocked by capacitors and let through an inductor.
AC is (ideally) passed by capacitors and blocked by inductors.

The above is a special case when component values are large and frequencies are moderately high. When component values are moderate and frequencies are moderate, you must be more specific with what you want to know.

Large signal analysis is concerned with the operation of macroscopic models of circuits, whereas small signal analysis is concerned with more subtle characteristics.

As an example, a silicon diode, with a large signal model, drops "about 0.7 volts at room temperature". What about when your signal is 20 mVpp AC? How does the diode act then? Id = Ise^(Vbe@ref current)/(k*t for temp) is for larger signals. Smaller signal analysis is carried out by linearizing the larger signal model's curve; basically, you zoom in to the point where the curvature seems to disappear and a straight line results.

I know that under large signal analysis, a capacitor becomes open, but under small, it becomes a short. My main question is what is an inductor under small signal analysis. Is it a short under large signal, but open under small signal?

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It seems possible that you have picked up a false impression from your studies of some simplified circuit transistor amplifier analysis methods. One approach to this involves the following processes:

"Large-signal analysis", which might be better described as bias (or working-point) calculations. Here capacitors are taken to be open-circuits. Inductors may be taken to be short-circuits for biasing purposes , but sometimes their DC resistance is significant and might need to be included.

"Small-signal analysis", where the working-point information is used to set up say H-parameter or hybrid-pi small-signal transistor models. Capacitors may be taken to be short-circuits, but at above beginners' level their reactances may also be considered as they can influence the frequency response. Similarly, during this kind of small signal analysis inductors may be modelled as open-circuits, as reactances, or at a more advanced level as complex impedances.

In a full large-signal analysis, as in a computer transient analysis of an oscillator circuit, the inductors and capacitors are not represented as shorts or open-circuits, but are actually represented in terms of differential equations.